Compressed Sensing and Matrix Completion with Constant Proportion of Corruptions (1104.1041v2)

Abstract: We improve existing results in the field of compressed sensing and matrix completion when sampled data may be grossly corrupted. We introduce three new theorems. 1) In compressed sensing, we show that if the m \times n sensing matrix has independent Gaussian entries, then one can recover a sparse signal x exactly by tractable \ell1 minimimization even if a positive fraction of the measurements are arbitrarily corrupted, provided the number of nonzero entries in x is O(m/(log(n/m) + 1)). 2) In the very general sensing model introduced in "A probabilistic and RIPless theory of compressed sensing" by Candes and Plan, and assuming a positive fraction of corrupted measurements, exact recovery still holds if the signal now has O(m/(log^2 n)) nonzero entries. 3) Finally, we prove that one can recover an n \times n low-rank matrix from m corrupted sampled entries by tractable optimization provided the rank is on the order of O(m/(n log^2 n)); again, this holds when there is a positive fraction of corrupted samples.

• The paper demonstrates that sparse signals can be exactly recovered from Gaussian measurements using ℓ1 minimization even with a constant fraction of corruptions.
• It extends the analysis to general sensing models, proving robust signal recovery in distributed systems under stricter sparsity conditions.
• The study further shows that convex optimization can accurately complete low-rank matrices despite pervasive corruptions, offering practical insights for signal processing applications.

Compressed Sensing and Matrix Completion with Constant Proportion of Corruptions

The paper authored by Xiaodong Li explores advancements in the field of compressed sensing (CS) and matrix completion, particularly in scenarios where sampled data undergo gross corruption. The primary objective of this research is to improve and extend previous results on signal and matrix recovery in the presence of significant corruptions, utilizing tractable optimization methods. This summary provides an overview of the main contributions, theoretical insights, and implications of the paper.

Core Contributions

The research introduces three significant theorems that address signal recovery from corrupted measurements:

1. Recovery in Compressed Sensing: The paper establishes that for a sensing matrix with independent Gaussian entries, a sparse signal can be accurately recovered using $\ell_1$ minimization even when a positive fraction of the measurements is corrupted. The sparsity condition here requires that the number of nonzero entries is proportional to $O(m/(\log (n/m)+1))$, where $m$ is the number of measurements, and $n$ is the signal dimension.
2. General Sensing Model: For a more general sensing matrix model, which accommodates distributed signal sensing akin to networked systems, exact recovery is achievable even with a fraction of corrupted measurements. The sparsity condition is slightly more stringent, demanding $O(m/(\log^2 n))$ nonzero entries.
3. Matrix Completion with Corruptions: The paper elucidates that a low-rank matrix can be recovered from sampled, corrupted entries via convex optimization. It posits that exact recovery is possible if the rank of the matrix is proportional to $O(m/(n\log^2 n))$, where $m$ denotes the number of samples and $n$ the matrix dimension. Again, this is contingent upon tolerating a positive fraction of sample corruptions.

Theoretical Implications

The work significantly extends the applicability of CS and matrix completion into realms where data integrity is compromised due to arbitrary or adversarial corruptions. It leverages sophisticated mathematical tools, including restricted isometry properties (RIP) and the golfing scheme technique to build robust recovery guarantees. The results imply error tolerance enhancements without compromising the tractability of optimization methods, providing optimized conditions for recovery that are superior to existing standards under equivalent settings.

Practical Significance

The implications of these results are profound for real-world applications such as signal processing, image reconstruction, and data sensing in unreliable network environments. They pave the way for resilient algorithmic designs capable of handling noise and corruptions in various applications ranging from sensor networks to medical imaging systems.

Future Directions

Future research may explore extending these techniques to broader classes of matrices or signal structures, including those prevalent in non-traditional senses like deep learning deployments. Further interdisciplinary efforts could also apply these optimization principles to enhance AI systems' robustness, particularly where data corruptions are prevalent.

Overall, the paper represents an important step in advancing our understanding and methodologies in signal and data recovery, addressing fundamental challenges within compressed sensing and matrix completion in the face of corruption.